Let's solve the problem step by step.

Problem:

The dimensions of the tank are:

• Length $L = 28 \, \text{cm}$
• Width $W = 24 \, \text{cm}$

Initially, the tank is filled with water up to a height of $h_1 = 3 \, \text{cm}$.

Additional water of volume $V = 6048 \, \text{cm}^3$ is poured into the tank, and the final water height reaches $\frac{3}{4}$ of the tank's total height ($H$).

We are tasked with finding the total height of the tank $H$.

Step 1: Write the volume equation

The volume of water initially in the tank is: $V_1 = L \cdot W \cdot h_1 = 28 \cdot 24 \cdot 3 = 2016 \, \text{cm}^3.$

The total volume of water after adding 6048 cm³ is: $V_{\text{total}} = V_1 + 6048 = 2016 + 6048 = 8064 \, \text{cm}^3.$

Step 2: Express the final water height

The final water height is $\frac{3}{4}H$, so the total volume is also: $V_{\text{total}} = L \cdot W \cdot \frac{3}{4}H.$

Substitute the known values of $L$, $W$, and $V_{\text{total}}$: $8064 = 28 \cdot 24 \cdot \frac{3}{4}H.$

Step 3: Solve for $H$

Simplify the equation: $8064 = 28 \cdot 24 \cdot \frac{3}{4}H = 2016 \cdot \frac{3}{4}H = 1512H.$

Solve for $H$: $H = \frac{8064}{1512} = 5.33 \, \text{cm}.$

Final Answer:

The height of the tank is $H = 5.33 \, \text{cm}$.

Let me know if you'd like further clarification! Here are some related questions for deeper understanding:

1. What is the formula for the volume of a rectangular prism?
2. How do proportions help solve problems involving fractions of a quantity?
3. What are other ways to measure the capacity of irregular containers?
4. How is volume conversion applied in real-life scenarios?
5. How do you calculate the amount of water needed to fill a tank completely?

Tip: Always double-check your calculations for consistency, especially when handling multiple steps.